applying Markov Chain switching model to Systemic Risk measures

(1)

0

EntropyIndexentropyIndexentr

opyIndexentropyIndexentropyI

ndexertySystemicriskmeasures

ystemikrisksystemicriskmeasur

esystemicrisksystemicriskmeas

ureMESpasdfghjklzxcvbnmqwer

tyuiopasdfghjklzxcvbnmqwerty

uiopasdfghjklzxcvbnmqwertyui

opasdfghjklzxcvbnmqwertyuiop

entropyIndexentropyIndexentr

opyIndexentropyIndexentropyI

ndexertySystemicriskmeasuree

ntropyshannonrényientropytsal

lisentropyentropyIndexentropy

IndexentropyIndexentropyInde

xentropyIndexertySystemicrisk

measureentropyshannonrényie

Markov Chain Switching

model applied on entropy

index based on Systemic

Risk Measures

MSc Econometrics Economics & Finance Ca’ Foscari University of Venice

Master thesis 2015

Student name: Imane EL idrissi Student number : 855749

(2)

1

Acknowledgements

The master‟s thesis in front of you is the result of months of labor and quite a bit of hurdles. Difficulties which I managed to overcome with the help of my supervisor Prof. Roberto Casarin. I would like to thank him ever so much for his patience and good advice. I found our meetings truly inspiring and he always gave me an energy boost to continue working and further improving my thesis.

I would also like to thank Prof. Monica Billio and prof. Eric Girardin for their encouraging words during my masters at Aix Marseille and Ca‟Focsari Universities.

Lastly, I am very grateful to have attended the MEEF program. The courses that I took challenged me and proved enormously helpful towards my professional career.

In addition, I am utterly grateful for the unquestionable love and moral support given to me by my family and some friends for their endless support and faith in me. Especially in times when I had to relieve my feelings, they were always there for me to listen and cheer me up.

Imane ELidrissi June 2015

(3)

2

Abstract

The aim of this thesis is to identify entropy regimes for the European financial market. I applied Markov switching model to the entropy series estimated on three different systemic risk measures, , Marginal Expected Shortfall (MES) and finally network connectedness In-Out measure. The approach used to estimate those models is Expectation Maximization. I considered two and three states. In order to select the model I used (AIC) and (BIC) criteria

Finally since the entropy measures exhibits a unit roots I investigate the role of the unit root in the detection of the entropy regimes.

Keywords: systemic risk; Systemic risk measure; , MES, Dynamic Causality,

network connectedness; Shannon entropy (1984); Rényi entropy (1960); Tsallis entropy (1988); Markov switching model; transition probability; AIC ; BIC; EM

(4)

3

Introduction

The financial crisis of 2007-2008 has pushed concerns about systemic risk and its measurement at the forefront of both academic research and supervisory policy agenda. In particular, ongoing work by the Basel Committee and the Financial Stability Board striving to set new regulatory requirements for Systemically Important Financial Institutions (SIFI) requires that an agreement can be reached on which characteristics make a financial institution more prone than others to be severely hit by system-wide shocks (systemic resilience or participation) or to propagate such shocks to other institutions, thereby amplifying their overall impact (systemic contribution)1.

New mathematical and computational tools have allowed researchers to work on undiscovered areas and to create new measures and approaches to systemic risk measurement. In particular, systemic risk analysis has improved a lot using extended versions of VaR based measures as , defined as the difference between the VaR of the financial system conditional on the institution being under distress and the VaR of the financial system conditional on state of that institution. Another relevant measure is the marginal expected shortfall (MES) proposed by Acharya et al. (2010), it considers the average return of each firm during the 5% worst days for the market. Additionally to these measures Diebold and Yilmaz (2009) introduced a new approach of connectedness which called In-Out measure, it is an international spillover index based on assessing shares of forecast error variation in different locations ( firms, markets, countries,etc) due to shocks arising elsewhere.

In the same way, few years ago new econometrics measures appeared considering the connectedness among financial institutions, one of the new measures is called Dynamic Causality index introduced by Gemantsky, Lo and Pelizzon and also (Billio et al 2012) which is capturing the degree of interconnectedness by looking at Principal Components and Granger Causality relations between returns and the international spillover index. Thus it is interesting to compare this new approach to understand the systemic risk with some previous categories of measures in order to identify the best risk measure that could capture possible missed elements of the contagion effect observed during financial crises and also to characterize network structure. These measures consider the financial system as a “portfolio” of institutions, where the shocks of the market price could impact the others. These measures are constructed primarily to capture the contribution to systemic risk of individual institutions. In particular, they capture the relation between the distress of each individual firm and the distress of the whole system.

Finally, my study relies on the use of entropy applied to a feature distribution estimated on the market, such as the cross-sectional systemic risk measures introduced above MES and network connectedness In-Out measure. Indeed, the change of entropy built on these measures could reveal signs of changes on systemic risk.

1_{Analytically, one may want to distinguish between situations where bank A reacts more than others to an}

exogenous shock and situations where Bank A is a source or an amplifyer of endogenous systemic events. Both dimensions of systemic importance are in practice clearly inter-related. The participation vs contribution approach was proposed by Drehman and Tarashev (2010).

(6)

5

Econometric tool model applied in this thesis is the Markov switching model, which is an important reference point since the work of Hamilton (1989, Econometrica). These models have been increasingly used in financial time-series econometrics thanks to their ability to capture some key features, such as heavy tails, volatility clustering, and mean reversion in asset returns [see Cecchetti and al. (1990), Pagan and Schwert (1990), Turner and al. (1989), Gray (1996), Hamilton (1988), and Timmermann (2000), among others]. In contrast to linear models those assume stationary distributions (such as ARIMA models), regime-switching models are based on a mixture of parametric distributions whose mixture probabilities depend on unobserved state variable(s). A key difference between the various regime-switching models lies in the stochastic structure of the state variables S. For instance, the state of the unobserved process can be modeled by a discrete time/discrete space Markov chain, which can has either fixed or time-varying transition probabilities, or by an independent stochastic state variable.

In this analysis I focus on univariate mean/intercept and variance switching models only considering the cases of two and three states with no time-varying transition probabilities. I compare three entropy indexes Shannon (1984), Rényi, (1960) and Tsallis (1988) applied on the three cross section distribution of risk measures mentioned before, ΔCOVAR, MES and In-Out.

These popular models are applied for the entropy indexes associated with each risk measure obtained from the available total number of financial institutions included in the European market for the period January 02nd 1986 to May 12th 2014, the data corresponding to the three entropy indexes. In order to compare the goodness of fit of the models we compute the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). In the models estimations I use the MS_Regress Matlab Tool-Box developed by Marcelo Perlin (Perlin, 2014) for estimating Markov Chain Switching Models (MCSM) through the maximum likelihood method and Hamilton´s filter.

This thesis is organized as follows, after this introduction, in section 1 and 2, I will offer a literature review covering past studies into systemic risk and systemic risk measures, which offer a suitable backdrop to the results of this paper. In section 3, I will discuss the entropy measures applied on the risk measures introduced before focusing on their differences and characterizations. I follow up by revising the approach of Markov Chain Switching Models and its importance in section 4. Before introducing and discussing empirical results in section 6 and 7 for data with unit root and without unit root, a detailed description of the estimation used in the empirical results is given on section 5. The intention of this is to give the reader a factsheet with which to study the empirical results of section 6 and 7. Finally, in section 8, I provide a conclusion of the outcomes of this study. As the study involves a substantial amount of empirical work, the main body of this thesis covers and discusses the most important elements thereof. The empirical appendix on section 9 offers detailed explanations about the MATLAB code implemented, general summary and statistics.

(7)

6

1. Systemic Risk

1.1 Definition and literature review.

There is no accord regarding the concept of financial stability and systemic risk. The materialization of systemic risk during the recent global financial crisis proved that the financial safety net and financial institutions significantly underestimated it. Systemic risk turned out to be much more than just the composition of individual types of risks affecting financial institutions. Whereas liquidity risk, credit risk, operational risk, etc. can be directly attributed to a given institution, systemic risk can only be attributed indirectly. Before the global financial crisis those types of risk were frequently considered separately. However, the interaction (correlation) between them leads to undesired and unexpected costs and consequences and when aggregated to systemic risk. Systemic risk evolves along with the development and growth of financial markets, regulations and collective behavior of market participants and it may be prompted by regulatory arbitrage. It is useful to explore the Systemic Risk definition. As Patrick Liedtke discusses, “what is truly remarkable is that at this point in time no definition of systemic risk exists that would be both fully convincing and generally accepted”

1. Clearly it represents an important aspect to deal with in order to identify an appropriate measure. The European Central Bank defines systemic risk as “the risk that the inability of one institution to meet its obligations when due will cause other institutions to be unable to meet their obligations when due. Such a failure may cause significant liquidity or credit problems and, as a result, could threaten the stability of or confidence in mar-kets”

2. In this sense, the Contagion Effect that characterizes financial crises could be associated with one of the mechanisms by which systemic risk is observed. Of course, a good measure of systemic risk has a relevant role for policy makers, as Acharya comments “the need for economic foundations for a systemic risk measure is more than an academic concern as regulators around the world consider how to reduce the risks and costs of systemic crises” 3. As a result, an appropriate systemic risk measure should include both theoretical and practical relevance.

There are at least three approaches that can be used to assess the build-up of systemic risk. The first focuses on monitoring traditional indicators of financial soundness or stability in order to assess broad developments in the financial system; the second focuses on measuring interconnectedness between financial institutions; and the third focuses on changes in the behavior of prices of financial assets.

(8)

7

2. Systemic Risk Measures

In this section I introduce three cross sectional distribution of marginal systemic risk measures first Marginal expected Shortfall (MES), Delta Conditional Value-at-Risk , and network connectedness called In-Out measure.

2.1 Systemic risk measure: Marginal Expected Shortfall (MES)

MES is a measure of the sensitivity of a financial firm to systemic risk .Systemic risk is defined as “the risk that the intermediation capacity of the entire financial system is impaired” (Adrian & Brunnermeier p.1). The first way to look at this risk is to examine the extent to which an individual bank or institution is affected by a systemic crisis. To this end, Marginal Expected Shortfall (MES) as introduced by Acharya et al. (2010) is considered. It is the average return of an individual institution during the 5% worst days of the market:

*

+

In which

= Marginal Expected Shortfall during the 5% worst trading days on the market

= Return of institution i

= The 5% worst outcomes at the market

MES measures the loss of an individual Institution when the entire market is doing poorly. A more negative value for MES indicates more systemic risk. From now on, a higher value of MES is interpreted as a more negative value for MES. Thus a higher value for MES implies more systemic risk.

2.2 Systemic risk measure: :

The second way to view systemic risk is to measure the contribution of an individual institution i to the overall systemic risk of the system. For this purpose, is used as discussed in Adrian and Brunnermeier (2011). CoVaR is the conditional Value at Risk, in which “Co” stands for conditional, contagion, or co-movement. CoVaR of the institution i relative to the system is the VAR of the whole financial sector conditional on institutions under distress. Furthermore, is the difference between the CoVaR of the financial system conditional on institution i being in distress and CoVaR of the financial system conditional on institution i operating in its median “normal” state, (Brunnermeier, Dong, and

(9)

8

Palia, 2012). As deduction CoVaR, captures the marginal contribution of a particular institution i (in a non-causal sense) to the overall systemic risk.

Denote as the contribution of Institution i to the systemic risk of the entire system at time t. Denote as the Value at Risk of the entire system conditional on bank i being in distress at time t. It is the q % Value at Risk of the entire system conditional on institution i operating at its VaR level. Denote as the Value at Risk of the entire system conditional on bank i operating at its median state at time t. As one can conclude from subscript t, the terms are time varying, which implies that the model is a dynamic conditional model instead of a stable unconditional model.

There are several advantages to the measure. First, while emphasizes on the contribution of each institution to overall system risk, traditional risk measures focus on the risk of individual institutions. Further, it is general enough to study the risk spillovers from institution to other across the whole financial network. Δ for example captures the increase in risk between two institutions, when the first falls into distress. To the extent that it is causal, it captures the risk spillover effects that institution i causes on institution j. Of course, it can be that institution i distress causes a large risk increase in institution j, while institution j causes almost no risk spillovers over institution i. The most common method to test for volatility spillover is to estimate a multivariate GARCH processes.

2.3 Systemic risk measure:

Systemic risk highlights on contagion, spillover effects and co-movement in financial markets, some measures of systemic risk are based on principal components analysis (PCA) or Granger-causality test. For instance absorption ratio based on PCA was used for systemic risk measurement as proposed by Kritzman et al. In the same vein, Diebold and Yilmaz (2014) propose several connectedness measures built from variance decompositions, and they argue that these piece of variance decomposition provides a natural and insightful measure of connectedness among financial asset returns and volatilities. They also show that variance decomposition defines weighted, directed networks, so that their connectedness measures are intimately-related to key measures of connectedness used in the network literature. „Building on these insights, we track both average and daily time-varying connectedness of major U.S. financial institutions' stock return volatilities in recent years, including during the financial crisis of 2007-2008. On the other hand, Billio et al. (2012) suggested several measures of systemic risk focused on the connectedness of the financial institutions to quantify the interrelationship between the monthly returns of hedge funds, banks, brokers and insurance

(10)

9

companies based on principal Component analysis (PCA) and Granger-causality tests. They introduced an indicator Number of Connections (NC) calculated from Granger-causality test to measure the degree of systemic risk. When applying PCA, the basic idea is that systemic risk is getting higher when the largest eigenvalue increases explaining most of variation of the data. When applying Granger-causality test. The first step is to estimate a Generalized Autoregressive Conditional Heteroskedasticity model for the returns using a structure GARCH (1, 1) and conditional on the system information that is developed over the sigma-algebra б:

So this structure allows identifying the relationships among institutions that are included in the network. Applying the Granger-causality test they work on the directionality of the relationships between the elements in the system. Now, they define the following indicator of causality:

(

And also when ( . This indicator is used to define the connections between N financial institutions. The degree of Granger causality (DGC) among all N (N-1) pairs of N financial institutions is given by the formula:

∑ ∑

As a result, when DGC exceeds a threshold K the systemic risk increases and it is possible to count the Number of Connections that affect a particular institution (IN) and are affected by the same institution (OUT). So, when systemic risk is raising the level of system interconnectedness also increases. Therefore, the total number of connections is given by:

∑

(11)

10

3. Entropy index

3.1 Introduction

“Entropy” this word appeared in 1865 when the German physicist Rudolf tried to rename the irreversible heat loss, what he previously called “equivalent-value”. This name was chosen for the reason that in Greek, en+tropein” means “content transformative” or “transformation content” .since this years the entropy has an important application in thermodynamics. Being defined as the sum of “heat supplied” divided by “temperature”, it is central to the Second Law of Thermodynamics. Using the Conrad definition “In our probabilistic context entropy is viewed as a measure of the information carried by the probability distribution, with higher entropy corresponding to less information (more uncertainty, or more of lack of information)”

Entropy can be used and measured in many other fields than thermodynamics. One of the important applications of entropy in information theory is Shannon entropy. In finance as well the application of entropy can be considered as the extension of the information entropy. It is an important tool in asset pricing and portfolio selection.

All the above papers recognize that entropy could be a good measure of risk; however it seems to be difficult to use this measure. In the systemic risk context considering the financial system behavior as random process where the values of the risk measure associated with each institution included in are realizations of this process, it is possible to estimate an entropy function for some specific period and then analyze the performance of this entropy index over the time. Thereby, comparison between entropy indexes associated with different risk measures could be a good method to understand systemic risk dynamics. My main motivation is to compare tree different entropies indexes, each one based on tree different systemic risk measures, , MES, In-Out in order to select the best model to identify and forecasts financial crises.

3.2 Concepts of Entropy Used in Finance 3.2.1 The Shannon Entropy (1984)

The Shannon entropy of a probability measure p on a finite set X is given by:

∑ (1)

(12)

11

When dealing with continuous probability distributions, a density function is evaluated at all values of the argument. Given a continuous probability distribution with a density function f(x), we can define its entropy as:

E = -∫ (2)

Where ∫ =1 and f(x) ≥0.

3.2.2 The Rényi (1960) and Tsallis (1988) Entropy

For any positive real number α. 0 < < ∞ The Rényi and Tsallis entropy of order , of a probability measure p on a finite set X is defined to be.

(∑ ) (3)

= ∑ (4)

We need 1 to avoid dividing by zero, but l‟Hôpital‟s rule shows that the Rényi entropy approaches the Shannon entropy as approaches 1

= ∑

Maszczyk and Duch (2008) showed that, the Shannon entropy is a special case of the Rényi and Tsallis entropies. Specifically, according to the value of α, the measures in Equations (3) and (4) attribute roughly weight to the tails of the distribution. At a first glance, the main difference between Shannon and Rényi entropies is the placement of the logarithm in the expression. In Shannon entropy (1), the probability mass function weights the log ( ) term, whereas in Rényi entropy the log is applied for the total summation that involves the power of the probability mass function. A large positive value implies this measure is more sensitive to events that occur often, while for a large negative value shows more sensitiveness to the events which happen seldom.

(13)

12

In order to compare further with Shannon entropy let me rewrite Rényi entropy as

= (∑ ) = - log (∑ )

= - log (∑ ) ₍₅₎

In (5) the probability mass function also weights a term that now is the power of the probability mass function. At a deeper level, Rényi entropy measure is much more flexible due to the parameter enabling several measurements of uncertainty (or dissimilarity) within a given distribution, the higher the and the less the distribution entropy far from the uniform. In other words the farther a distribution is from the uniform, the thinner its tails are. While, the Tsallis entropy assigns less importance to randomness that is it penalizes uniformity in the distribution. This entropy behaves in the same way of Shannon; there is only difference in the magnitude. When increases or decreases the magnitude also increases or decreases.

4. Markov Chain Switching Regime

4.1 Glossary

 Regime-Switching Model

Is a parametric model of a time series in which parameters are allowed to take on different values in each of given fixed number of regimes

 Markov Chain (MC)

Is a process that consists of a finite number of regimes, where the probability of moving to a future regime conditional on the present regime is independent of past regime

 Markov-Switching Model (MSM)

Is a regime switching model in which the shifts between the states evolve depending on an unobserved MC.

 Regime-Switching Model

Is a parametric model in which parameters are allowed to take on different values in each of given fixed number of regimes (states).

(14)

13

 Transition Probability

The probability of switching from state j to state i

 Filtered Probability of a Regime

The probability that the unobserved Markov Chain (MC) for a Markov-switching model is in a particular regime in period t, conditional on observing sample information up to period T.

 Smoothed Probability of a Regime

The probability that the unobserved MC for a Markov-switching model is in a particular regime in period T, conditional on observing all sample information.

4.2 Definition and Importance of Regime switching models

Regime-switching models have been discussed more than 50 years ago. Early econometrics focused on a simple model that incorporates a single non-recurring structural break. These time-series models in which parameters are allowed to take different values in each fixed number of “regimes.” A stochastic process supposed to have generated the regime shifts is included as part of the model, that allows for model-based forecasts which incorporate the possibility of future regime shifts. In some special situations the regime in operation at any point in time is directly observable. Generally the researchers must conduct inference about which regime the process was in at past points in time.

Describing changes in the dynamic behavior of macroeconomic and financial time series has been the main purpose of using these models in the applied econometrics literature. Regime-switching models could be divided into two categories, “threshold” models and “Markov-switching” models, which are used in my thesis. The principal difference between these approaches is in how the advancement of the state process is modeled. The first models, introduced by Tong (1983), suppose that regime shifts are triggered by the level of observed variables in relation to an unobserved threshold. The second models, introduced to econometrics by Goldfeld and Quandt (1973), Hamilton (1989), suppose that the regime shifts evolve depending on a Markov chain. Regime-switching models became increasingly popular tool of modeling for applied work, these models provided an alternative and important approach to understand and interpret how an economy‟s growth rate changes over time. An important development in this regard is the Hamilton (1989) to model and identify the phases of the US business cycle. There is other application which includes modeling regime shifts in time series of inflation and interest rates (Garcia and Perron, 1996).High and low volatility regimes in equity returns (Hamilton and Susmel, 1994; Hamilton and Lin, 1996) and time

(15)

14

variation in the response of economic output to monetary policy actions (Lo and Piger, 2005; Kaufmann, 2002).

4.3 Markov chain Switching

There is huge interest in modeling dynamic behavior of macroeconomic and financial time series. One of the challenges for this analysis is that these time series usually undergo changes in their behavior over reasonably long sample periods. The change usually occurs in the form of a “structural break”, in which there is a shift in the behavior of the time series due to some permanent change in the economy‟s structure. The change in behavior might also be temporary, as in the case of wars or “pathological” macroeconomic episodes such as economic depressions, hyperinflations, or financial crises. These shifts might be both temporary and recurrent, in that the behavior of the time series might cycle between regimes. For example, as indicated by early studies of the business cycle, the behavior of economic variables changed dramatically in expansions vs. recessions.

The constant parameter time series models might be inadequate to describe the potential for shifts in the behavior of economic time series and their evolution. As a result, recent decades have seen extensive interest in econometric models designed to incorporate parameter variation. One approach to describing this variation, denoted a “regime-switching” model in the following, is to allow the parameters of the model to take on different values in each of some fixed number of regimes, where, in general, the regime in operation at any point in time is unobserved by the econometrician. The process that determines the arrival of new regimes is assumed known, and is incorporated into the stochastic structure of the model which allows the econometrician to draw inference about the regime that is in operation at any point in time, as well as form forecasts of which regimes are most likely in the future.

Quandt (1958) first tried to estimate the parameters of this linear regression system; he supposed that such a system follows two separate regimes. Similar to the model in (1) and (2)

= + ∼ N (0, ) (1) = + ∼ N (0, ) (2)

Denote as return series, and and define the mean and variance in period i (further > or < is assumed). He assumed a point in time at which the system switches the regime and the data is described by a different regression equation. Using maximum likelihood estimation, Quandt was capable to infer the corresponding turning point and to

(16)

15

determine the regression parameters. Despite, his model considers only the possibility of a single switch and it was only tested in a hypothetical sampling experiment.

To allow studies of multiple regime-switches Quandt (1972) prolonged his earlier model and applied it to the US housing market between June 1959 and November 1969, by introducing the λ-method, where λ and1 – λ are unknown probabilities for the observed data points being driven by the regime 1 or 2, as result he got a significant results for the two regime.

Goldfeld and Quandt (1973) introduced the first Markov-switching model after relaxing the assumption of constant regime probabilities. They assumed that there is dependence of the current regime on its preceding state through a Markov chain process. To capture the state dependency they introduced a transition probability matrix, which governs the transitions across states. When they applied τ -method to the same data sample as Quandt (1972) they obtained also similar results as for the λ-method. All these applied works shows that these models became more sophisticated and complex because of the data availability and computational improvements..

5. Estimation

A general Markov Switching model can be estimated with two different ways first by Maximum likelihood or by Bayesian inference (Gibbs-Sampling). This thesis focuses and explains the first method.

This estimation precedes a recursive filter and numerical maximization by a method by an EM algorithm, which is appropriate for maximizing likelihood with unobserved variables or missing observations. I describe these procedures in this section.

5.1 Specification

I define the model structure that I will apply in this estimation. I am working on intercept and variance switching models, for two-states the model is given by:

{

Where represents one entropy index for each of the risk measure in the period t and and represent its mean and variance. I assume that and follow a distribution that depends on a latent process (variable) , this variable follows a Markov Chain process. At

(17)

16

each point in time, the process is in one of two regimes, which we Indicate by and .

In both regimes, the entropy ( follows a normal distribution, though with different means and different variances. I use the function f to denote the normal probability density function.

√ (

)

And the log likelihood of this model is given by:

∑ (

√ ( ))

(1)

If all of the states of the world were know, that is, the values of are available, then estimating the model by maximum likelihood is straightforward. All I need is to maximize equation (1) as a function of parameters , , and . But is not the case of a Markov Switching model, because the states of the world are unknown. So in this case it is necessary to change the notation of the likelihood function. By considering as the likelihood function for state j conditional on a set of parameters ( ), then the full log likelihood function of the model is given in the equation (2)

∑ ) (2)

The latent process follows a first order Markov Chain, this means that the probability for regime 1 to occur at time t only depends on the regime at time t-1 . I denote these Transition probabilities by

= Pr [ = i | = j]

The term represents the probability of switching from state j to state i. The transition probabilities for the departure states j should add up to one, i.e., + = 1 and + = 1. So, for a binary process , I have two free parameters. and .The transition matrix is defined by:

(18)

17 P = (

) = (

)

To estimate the parameters of the regime switching models using a maximum likelihood approach. As with other conditional models such as ARMA- or GARCH-models, the likelihood function will take a conditional form, too. I gather the parameters of the model in a vector = ( , )

5.2 The Hamilton filter

In order to maximize the Log-Likelihood function is necessary to estimate using the Hamilton‟s filter that allows estimating these probabilities using the available information in the period t-1 denoted by . In this sense, the Log-Likelihood function will be estimated as a function of the parameters and the available information, then we have:

∑

Four steps are needed to realize the algorithm:

1. Establish a guess of stating probabilities in t=0 of each state, for j=1,2 which satisfy the relation

2. Prediction of regime probabilities i.e. In t=1 compute the probabilities for each state given and the transition matrix, where

∑

(19)

18

3. Calculation of filtered probabilities through update i.e Update the probability of each state with the new information from time t. This can be done using the parameters of the model in each state, and the transition probabilities and for computing the likelihood function in each state over the period t. Next step, update the probability of each state given the new information as follows:

∑

4. Now move to t+1 and repeat the steps 2 and 3 until t=T, completing all the available data in the T periods. As a result, we obtain all the state probabilities across the sample.

5.3 Maximization

After computing the states probabilities by applying Hamilton‟s filter, it is possible to maximize the log likelihood function using a numerical method. In this case the numerical method used in MS_Regress_fit.m consists on applying four algorithms included in fmincon function in MATLAB.

1. SQP Algorithm

2. Interio-point Algorithm 3. Active-set Algorithm

(20)

19

6. Results

6.1 data

I consider the daily closing price for the stocks of the European financial institutions from 2nd January 1986 to 12th may 2014. The data has 7398 observations. According to the dynamics of system the number of institutions is changing across the time, beginning with 30 institutions and ending with 203.

To estimate systemic risk measures I used a rolling window approach (e.g., see Zivot and Wang J.(2003), Billio et al.(2012), Diebold and Yilmaz(2014)) with a window size of 252 daily observations, which corresponds approximately to a year of daily observations.

6.2 Descriptive statistics

In this section I study the mean and variance behavior of each entropy index related with each risk measure , MES and INOUT, in order to identify different regimes of entropy and multimodal entropy distribution.

1.

Figure 1: The entropy indexes for

0 1000 2000 3000 4000 5000 6000 7000 8000 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 dCoVar Period v a lu e s Entropy Shannon Entropy Rényi Entropy Tsallis

(21)

20

Figure 2: The entropies distributions for measure

Figure 3: First difference entropies indexes for measure

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0 500 1000

2 standard deviations Shannon Entropy distribution on dCoVar

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

0 500 1000

2 standard deviations Rényi Entropy distribution on dCoVar

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0 500 1000

2 standard deviations Tsallis Entropy distribution on dCoVar

7.24 7.26 7.28 7.3 7.32 7.34 7.36 x 105 -0.2 0 0.2 firstdiff Shannon-dCoVar 7.24 7.26 7.28 7.3 7.32 7.34 7.36 x 105 -0.5 0 0.5 firstdiff Rényi-dCoVar 7.24 7.26 7.28 7.3 7.32 7.34 7.36 x 105 -0.5 0 0.5 firstdiff Tsallis-dCoVar

(22)

21

For the cross-sectional systemic risk measure , the Figure 1 shows the plots of the three entropies Shannon, Tsallis and Rényi. The two first entropies behave in the same way, the difference remains in the magnitude. The Rényi behaves differently. Using this entropy could change totally the estimation of MSCM, specifically the presence of the states. I will have more information by analyzing the histograms and the first difference included in the Figure 2 and 3.

In order to get some conclusion about the state of the mean and of the variance, as the histograms show the mean is not representative because the entire histograms exhibit clearly multimode, so taking the mean like a normal model is not working properly for this data. Therefore I need to use models which account for multimodality such as mixture normal or Markov Switching, are more suitable, for these data.

Even more, the first difference series shows the presence of at least two states for the variance behavior. As a result, taking in account these facts I consider that a three states switching Markov chain could be the best model to approximate the entropies indexes. I will compare these expectations with the estimation results.

At the end my expectation for the entropy related to this systemic risk measure, is a Markov Chain Switching Model three states for the Rényi entropy, and two states for the Shannon and Tsallis.

2. Marginal Expected Shortfall (MES)

Figure 4: The entropy indexes for MES

0 1000 2000 3000 4000 5000 6000 7000 8000 0.4 0.5 0.6 0.7 0.8 0.9 1 MES Period v a lu e s Entropy Shannon Entropy Rényi Entropy Tsallis

(23)

22

Figure 5: The entropies distributions for MES measure

Figure 6: First difference entropies indexes for MES measure

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0

500 1000

2 standard deviations Shannon Entropy distribution on MES

0.7 0.75 0.8 0.85 0.9 0.95 1

0 500 1000

2 standard deviations Rényi Entropy distribution on MES

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0

500 1000

2 standard deviations Tsallis Entropy distribution on MES

7.24 7.26 7.28 7.3 7.32 7.34 7.36 x 105 -0.1 0 0.1 firstdiff Shannon-MES 7.24 7.26 7.28 7.3 7.32 7.34 7.36 x 105 -0.1 0 0.1 firstdiff Rényi-MES 7.24 7.26 7.28 7.3 7.32 7.34 7.36 x 105 -0.1 0 0.1 firstdiff Tsallis-MES

(24)

23

From the Figure 4 I have the same conclusions for this measure, the Shannon and Tsallis entropies are similar in the behavior. Furthermore, after the period 3000 the mean increases and also the behavior becomes more volatile. Unlike the Rényi „s entropy, the histograms of Marginal expected shortfall reflect that it could be a Markov chain switching model with two state for the Shannon and three states for the Tsallis and Rényi In addition, the first difference shows that there are at least two states for the variance.

As a result my expectation for the entropy related to MES is a Markov Chain Switching model three states for all the entropies.

3. In-Out

Figure 7: The entropy indexes for In-Out

0 1000 2000 3000 4000 5000 6000 7000 8000 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 INOUT Period v a lu e s Entropy Shannon Entropy Rényi Entropy Tsallis

(25)

24

Figure 8: The entropies distributions for In-Out measure

Figure 9: First difference entropies indexes for In-Out measure

The entropies of In-Out measure are increasing with time and tending to the Rényi entropy, as result I have bigger variance and a higher mean over the time. But for the histograms I remark a clearly a different behavior for the Rényi entropy with a bigger concentration around the

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 500 1000

2 standard deviations Shannon Entropy distribution on INOUT

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

0 1000 2000

2 standard deviations Rényi Entropy distribution on INOUT

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 500 1000

2 standard deviations Tsallis Entropy distribution on INOUT

7.24 7.26 7.28 7.3 7.32 7.34 7.36 x 105 -0.1 0 0.1 firstdiff Shannon-INOUT 7.24 7.26 7.28 7.3 7.32 7.34 7.36 x 105 -0.2 0 0.2 firstdiff Rényi-INOUT 7.24 7.26 7.28 7.3 7.32 7.34 7.36 x 105 -0.1 0 0.1 firstdiff Tsallis-INOUT

(26)

25

mode .however it is possible to have three states for all of the entropies. In addition the first difference for the Shannon entropy is more even, in this case it will be difficult to identify the states but for the other entropies it has at least to states for MCSM.

Additionally when I plot the normalized entropy against the non-normalized, I remark that for the Shannon the normalization does not affect the pattern, I have only the change in the scales, because this entropy was normalized by using this formula (1) where k is the bins. However checking the magnitude the entropies before and after the normalization the Rényi entropy becomes the highest one, it means that the normalization has a bigger effect over Shannon and Tsallis than Rényi.

Entropy Shannon_n = (1)

The expected result for the entropy based on In-Out systemic risk measure is three states for all the entropies.

6.3 Estimation results

I apply Markov chain switching model to entropy indexes, using the estimation method outlined in the previous section, for two states and three states. The data consists of three entropy indexes each one associated to one of these risk measures , MES and In-Out for the period mentioned before for two cases non-normalized and normalized.

The thesis includes the estimation of 36 models. In order to compare the fitness of the models I compute the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC).

AIC = -2 ln (L) + 2 ln (k) BIC = -2 ln (L) + k ln (n)

Where L is the likelihood evaluated at the mode of the parameters, n the number of the observations and k is the number of parameters in the model. BIC compares the (negative) likelihood, but penalizes for increased model complexity. For small sample sizes, BIC tends to choose less complex models, but the series in the empirical study have many observations, so this is not an issue.

After checking the estimation results I can identify some issues. Table 1 summarizes the log likelihood values evaluated at the parameter modes obtained for the different states. As expected, in most of the cases the log likelihood increases with the number of states in the

(27)

26

MCSM, with a big jump for the MES. According to the BIC values, a switching model with three states is the strongest candidate. I examine the results from these models further.

Tables 2-3 and 4 contain the values of (BIC) and (AIC) estimated for each entropy in two and three states MCSM. First of all I focus on the normalized case, for the Shannon and the Tsallis entropy, the best measure is for three states, unlike the Rényi entropy the best one is MES also in three states. Similarly the non-normalized indexes have the same results.

One the other hand when I compare the criteria for the normalized and non-normalized indexes it is clear that the models for the normalized indexes show a better performance than the non-normalized. In fact sometimes the difference between both is 29 times for the Shannon and Tsallis entropies. However in the Rényi case there are no significant differences, this could be associated to its own formula. As result I can say that the normalization is a good tool to improve the quality of the estimation in the Shannon and Tsallis entropies, but not in the case of the Rényi entropy.

When I analyze the models associated to In-Out measure, for all the entropies the best result is the three states and also the normalized case is better than the non-normalized. As result MCSM three states for normalized Rényi entropy index based on this measure has the lowest value for both criteria.

It is important to underline that when I compare the models related to Shannon entropy, the ratio for the (AIC) between the best model and the worst is 4.42 times, 1.43 times for the Rényi and 4.08 times for the Tsallis, as a conclusion is less sensible to the use of different risk measures, in other words this entropy has a lower model risk unlike the other entropies. Last important remark, this ratio is very high in the case of Shannon entropy for the non-normalized data (17.02) times, in the same why, by analyzing the (BIC) criteria I got similar results. In the case of the Shannon and Tsallis the normalization has a positive effect in reducing model risk but no a significant results for the Rényi.

Table 1: The likelihood evaluated at the parameter modes obtained for different states

logliklihood non-Norm Norm non-Norm Norm non-Norm Norm

∆CoVaR -8306.2301 5701.4137 10723.1721 13060.9947 -5737.2494 5559.2982 MES -2040.9473 11966.2589 13994.4332 13651.2607 436.631 11733.1787 In-Out -10821.0986 3186.5452 11522.4526 13860.2752 -8083.1864 3213.3613 ∆CoVaR -623.6275 13384.0163 10723.0509 13061.0641 1784.8115 13081.3592 MES -1078.0593 12929.5845 16378.6258 18716.4484 436.6299 11733.1776 In-Out -4903.11 9104.5338 14295.3194 16633.142 -1095.6283 10200.9194 2 S 3 S

Tsallis entropy Index Rényi entropy Index Shannon entropy Index

(28)

27

Table 2: (BIC) and (AIC) values for the Shannon entropy index in two and three states

Table 3: (BIC) and (AIC) values for the Tsallis entropy index in two and three states

Table 4: (BIC) and (AIC) values for the Rényi entropy index in two and three states

After the estimation of the models, comparing these results with my expectations, I got the same conclusions for all the entropies based on In-Out measure; it means that the best model is MCSM with three states. For the Rényi and Shannon entropy based on MES measure the efficient models are three states MCSM, which agrees with my expectation, additionally the model related to Tsallis index is a two state MCSM which is opposite to my beliefs. On the other side for all the entropies related to measure, the expectations are opposite to the results, notice that for the Rényi index the difference between the states is very small. Now it is relevant to analyze the results of the best model of each entropy index in the normalized case in terms of the switching process and the level of the states.

BIC -1.1349e+04 BIC -2.3879e+04 BIC -6.3196e+03

AIC -1.1391e+04 AIC -2.3921e+04 AIC -6.3611e+03

BIC 1.6666e+04 BIC 4.1353e+03 BIC 2.1696e+04

AIC 1.6624e+04 AIC 4.0939e+03 AIC 2.1654e+04

BIC -2.6661e+04 BIC -2.5752e+04 BIC -1.8102e+04

AIC -2.6744e+04 AIC -2.5835e+04 AIC -1.8185e+04

BIC 1.3542e+03 BIC 2.2630e+03 BIC 9.9131e+03

AIC 1.2713e+03 AIC 2.1801e+03 AIC 9.8302e+03

∆CoVaR MES In-Out

NORM Non_NORM NORM Non_NORM 2 S NORM Non_NORM NORM Non_NORM 3 S NORM Non_NORM NORM Non_NORM

BIC -2.6069e+04 BIC -2.7249e+04 BIC -2.7667e+04

AIC -2.6110e+04 AIC -2.7291e+04 AIC -2.7709e+04

BIC -2.1393e+04 BIC -2.7935e+04 BIC -2.2991e+04

AIC -2.1434e+04 AIC -2.7977e+04 AIC -2.3033e+04

BIC -2.6015e+04 BIC -3.7326e+04 BIC -3.3159e+04

AIC -2.6098e+04 AIC -3.7409e+04 AIC -3.3242e+04

BIC -2.1339e+04 BIC -3.2650e+04 BIC -2.8484e+04

AIC -2.1422e+04 AIC -3.2733e+04 AIC -2.8567e+04

2 S

NORM NORM NORM

Non_NORM Non_NORM Non_NORM

3 S

BIC -1.1065e+04 BIC -2.3413e+04 BIC -6.3733e+03

AIC -1.1107e+04 AIC -2.3454e+04 AIC -6.4147e+03

BIC 1.1528e+04 BIC -819.8082 BIC 1.6220e+04

AIC 1.1486e+04 AIC -861.2620 AIC 1.6178e+04

BIC -2.6056e+04 BIC -2.3359e+04 BIC -2.0295e+04

AIC -2.6139e+04 AIC -2.3442e+04 AIC -2.0378e+04

BIC -3.4627e+03 BIC -766.3522 BIC 2.2982e+03

AIC -3.5456e+03 AIC -849.2598 AIC 2.2153e+03

2 S

3 S

(29)

28

The transition matrix of the Shannon entropy based on has1´s in all the diagonal showing than there is a strong evidence of switching between the three states. However, the transition matrix has zeros out of the diagonal, showing a low efficiency of the Markov Chain process, which implies that there is no relation between the state in one period and the state of the next period. This is the transition matrix associated with this model:

P = ( ) Figure 10: Smoothed states Probabilities for Shannon entropy based on

0 1000 2000 3000 4000 5000 6000 7000 8000 0 0.5 1 Explained Variable #1 0 1000 2000 3000 4000 5000 6000 7000 8000 0.4 0.6 0.8

conditional mean of equation

0 1000 2000 3000 4000 5000 6000 7000 8000 0.02 0.04 0.06 Conditional Std of Equation #1 0 1000 2000 3000 4000 5000 6000 7000 8000 0 1 2 Time S m o o th e d S ta te s P ro b a b ili ti e s State 1 State 2 State 3

(30)

29

Figure 11: Conditional mean of Shannon entropy based on

Even more the level of the entropy is higher in the state three with 0.6992 followed by state one 0.5484 and state two 0.4267. In particular the state three has two relevant period of crises especially around the observations 4000 (crises of 2001) and before the period 6000 (crises of 2008), for this model the crises finish in 2013.in this case I can say that the state two reflects no crises period and the state one could be considered as the bullish market before the crises. Finally it is important to remark that the expected duration for the state three is 1.4 years. As regards the analysis of conditional mean related to this entropy index, the order is reverse for the lowest and mid-level because the lowest level of standard deviation corresponds to mid-level of entropy. And lowest level of entropy correspond midlevel of standard deviation, but when the level of entropy is high also the level of Standard deviation is high, when both are high the probability of systemic risk is high and that corresponds to period crises in 2001 and 2008 (see the figure 11), while the state one reflect in the case the period before the crises. Table 5 summarizes for each state the level and the value of both conditional mean and conditional standard deviation.

Table 5: levels of both Cond-mean and the Cond-standard deviation with their values

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0186 1088 0892 0901 1007

Cond-mean levels Cond-std levels

state1 0.5484 M 0.000988 L

state2 0.4267 L 0.001533 M

(31)

30

For the Rényi entropy based on MES measure I also obtain some interesting results. , the three-state model has values different of zero in three of the six transitions probabilities out of the diagonal. However, the probabilities related to transition from the first and the second states to the third state are equal or close to zero indicating low relation between the states for two different periods. But the probability of switching from state three to state two is 15%, so there is a significant relation between these states, this is the transition matrix:

P = (

)

Figure 12: Smoothed states Probabilities for Rényi entropy based on MES

0 1000 2000 3000 4000 5000 6000 7000 8000 0.7 0.8 0.9 Explained Variable #1 0 1000 2000 3000 4000 5000 6000 7000 8000 0.8 0.9 1

0 1000 2000 3000 4000 5000 6000 7000 8000 0.01 0.02 0.03 Conditional Std of Equation #1 0 1000 2000 3000 4000 5000 6000 7000 8000 0 1 2 Time S m o o th e d S ta te s P ro b a b ili ti e s State 1 State 2 State 3

(32)

31

Figure 13: Conditional mean of Rényi entropy based on MES

Here the level of the entropy is increasing from the state two to state one as follow, 0.8083, 0.8636, and 0.9167. This model it does not recognize the crisis period because it includes the crises before the 4000 and 5000 period and after 1000 period as a conclusion I have a lot of small crises with higher frequency, and in particular this model it doesn‟t recognize the crises of 2008. Simultaneously the expected duration for this state is 5723.12 (23years).

Following the idea of analyzing the level of entropy in this case the lowest level of conditional mean corresponds to high level of standard deviation, high conditional mean corresponds to mid-level of standard deviation and finally mid-level of conditional mean corresponds to lowest level of standard deviation. The state one has the highest level of entropy and it appears in these years: at the end of 1987, after May 2001 and also after May 2008 but it last till 2013 with small jump to state three in 2009 which lasts a few months.

0.8 0.82 0.84 0.86 0.88 0.9 0.92 0186 1287 0793 0194 1097 0501 0506 0713

state1 0.9167 H 0.000363 M

state2 0.8083 L 0.000563 H

(33)

32

For the In-Out measure the model with the best performance was the Rényi with a three states, for this case I have a probability of 3% to switch from state two to one and also for switching from three to two, I have no significant result between these two states and the state one.as show the transition matrix

P = (

)

Figure 14: Smoothed states Probabilities for Rényi entropy based on In-Out

For this analysis I have a decrease of the entropy level from the state one to three, with 0.9378 for the state one and 0.8367 for state two and finally 0.6893 for the state three.in this case I have only the state one from 3000 period till the end with the appearance of small switches between the state two and state one, this model does not recognize properly the crises especially the crises of 2008. This model could be showing that the connectedness of the

0 1000 2000 3000 4000 5000 6000 7000 8000 0.6 0.8 1 Explained Variable #1 0 1000 2000 3000 4000 5000 6000 7000 8000 0.7 0.8 0.9

0 1000 2000 3000 4000 5000 6000 7000 8000 0 0.02 0.04 Conditional Std of Equation #1 0 1000 2000 3000 4000 5000 6000 7000 8000 0 1 2 Time S m o o th e d S ta te s P ro b a b ili ti e s State 1 State 2 State 3

(34)

33

system has increased across the time making more difficulties to identify the crises using this risk measure.

Figure 15: Conditional mean of Rényi entropy based on In-Out

For the last model the level of entropy is different than the previous models because the lowest and the mid-levels of the conditional mean corresponds in the same order the high and the mid-levels of the conditional standard deviation, but as shown in the Figure 15 the difference between this two levels is not significant, as regard the high level of entropy it is clearly that the standard deviation has the lowest level. In this model since April 2001 till the end of the series the state one covers this period which recognizes the dot-com bubbles and the global financial crises. Additional remark in this period there is a jump but no change in the regime that could be possible because the conditional mean is not the value of the mean in the regime but is an average with weights given the smoothed probability, consequently the probability of the state one is changing but it continue to be the highest of the states.

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0186 0401

state1 0.7338 H 0.002239 L

state2 0.6325 M 0.000292 M

(35)

34

As a conclusion of this section I identify the ∆CoVaR three states Shannon entropy model as the best one for the identification of the crises, in spite of the relation between the states is not a strong as it is reflected in its transition matrix. On the other hand the MES and In-Out measure were not able to identify both properly the global financial crises even if the transition matrix shows a significant switching, however for the In-Out the switching process could be affected by the presence of positive trend of the connectedness of the system, thus it could be interesting to estimate the models after removing the trend.

7. MSCM Removing the trend

7.1 Tests

Let be the observed time series which contain a unit root, in order to check the existence of these roots, I will apply two tests:

The ADF test and the PP test the following way to proceed can be determined beforehand:

 I Apply the Phillips-Perron (PP) which assesses the null hypothesis of a unit root in a univariate time series, Values of h equal to 1 indicate rejection of theunit-root null in favor of the alternative model. A value of h equal to 0 indicates a failure to reject the unit-root null.

 I Apply the Augmented Dickey Fuller (ADF) test to check the null hypothesis of unit root existence. If the null hypothesis is rejected, the conclusion is there is no unit root (stationary),

After the ADF test I got h = 0 for the all entropies with a pValue around 0.56 which indicate a failure to reject the null hypothesis, it means a failure to reject the existence of unit root. Similarly the PP rejects null hypothesis h = 0, as result all the entropies have a unit root. (See table 1)

In conclusion as all the entropies indexes are not stationary, all of them have at least two states in the Markov Chain process. In order to identify clearly the states I removed the trend by running a linear regression of each entropy index against the time and I applied the MCSM on the residuals of this regression, so this procedure will not affect the states of the original index as it is not changing the non-stationary pattern

(36)

35

Table 8: ADF and PP tests on entropies indexes before removing the trend

7.2 Descriptive statistics

The residuals look much better after removing the trend. Figures16-18 and Table 6 show the residuals for the all entropy indexes and the results of the Tests after removing the trend. First of all the indexes are moving around zero, also there is a change for the Rényi entropy that now behaves similar to the others unlike the descriptive analysis seen above.

Figure 16: Residuals of entropy indexes based on CoVaR

h_ADFtest pValue h-PPtest pValue

Shannon entropy 0 0.5147 0 0.5147 Rényi entropy 0 0.5720 0 0.5720 Tsallis entropy 0 0.4778 0 0.4778 Shannon entropy 0 0.5877 0 0.5767 Rényi entropy 0 0.6506 0 0.6033 Tsallis entropy 0 0.5580 0 0.5580 Shannon entropy 0 0.5767 0 0.5767 Rényi entropy 0 0.6033 0 0.6033 Tsallis entropy 0 0.4863 0 0.4863

before removing the trend ∆CoVaR MES In-Out 0 1000 2000 3000 4000 5000 6000 7000 8000 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

residuals dCoVaR entropy index shannon entropy

Rényi entropy Tsallis entropy

(37)

36

Figure 17: Residuals of entropy indexes based on MES

Figure 18: Residuals of entropy indexes based on In-Out

0 1000 2000 3000 4000 5000 6000 7000 8000 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

residuals MES entropy index

shannon entropy Rényi entropy Tsallis entropy 0 1000 2000 3000 4000 5000 6000 7000 8000 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

residuals entropy index

shannon entropy Rényi entropy Tsallis entropy

(38)

37

Table 9: ADF and PP tests on entropies indexes after removing the trend

As result the detrend has a positive effect on the entropies related In-Out measure in the sense that the series behaves with more stability that allows to identify the mean states, additionally the Tsallis is more volatile than the others.

7.3 Estimation Result

Knowing that the normalization increases the quality of the estimation I will just focus on the normalized indexes. Using the criteria (BIC) and (AIC) to compare the efficiency the models I got the following results:

For Shannon and Tsallis entropies based of In-Out measure in three states were the best models, for the Rényi entropy based on MES still the best model as the previous case ( trend included). In this sense I can say that the detrend method is a good tool to clean the In-Out measure in order to increase its efficiency detecting the financial crises using the connectedness approach. Tables 7, 8 and 9 show the results.

Table 10: (BIC) and (AIC) values for the residuals of Shannon entropy index in two and three states

ADFtest pValue PPtest pValue Shannon entropy 1 1.0e-03 1 1.0e-03 Rényi entropy 1 1.0e-03 1 1.0e-03 Tsallis entropy 1 1.0e-03 1 1.0e-03 Shannon entropy 1 0.0036 1 0.0036 Rényi entropy 1 0.0010 1 0.0010 Tsallis entropy 1 0.0015 1 0.0015 Shannon entropy 1 1.0e-03 1 1.0e-03

Rényi entropy 1 1.0e-03 1 1.0e-03 Tsallis entropy 1 1.0e-03 1 1.0e-03 In-Out

after removing the trend

MES ∆CoVaR

NORM ALIZED M EASURE BIC AIC

∆CoVaR -19500 -19541 M ES -24485 -24526 In-Out -25611 -25653 ∆CoVaR -20586 -20669 M ES -26239 -26322 In-Out -26537 -26620 Shannon Entropy 2 states 3 states Shannon Entropy

(39)

38

Table 11: (BIC) and (AIC) values for the residuals of Rényi entropy index in two and three states

Table 12: (BIC) and (AIC) values for the residuals of Tsallis entropy index in two and three states

Now I will be focus on the results of the best model of each entropy indexes. I start by analyzing the Shannon entropy

P = (

)

The transition matrix in this case has a probability of 1% of switching from the state three to one and also of switching from state three to state one, but the other probabilities are null except the diagonal. Even more the smoothed states probability shows the presence of blocks of blank, in this case it is difficult to identify which state represent the crises. The state two shows the highest level of the standard deviation which corresponds to the lowest level of entropy, the state one in this model represents the highest regime. (See table 13 and Figure 13). Between April 2006 and January 2008 the switching was only between state one and three, this happened also between September 2008 and May 2010, the period of financial

∆CoVaR -26045 -26086 M ES -32441 -32483 In-Out -30719 -30761 ∆CoVaR -27461 -27544 M ES -33639 -33722 In-Out -30666 -30749

2 states Renyi Entropy

3 states Renyi Entropy

∆CoVaR -19019 -19061 M ES -23848 -23890 In-Out -25290 -25331 ∆CoVaR -20076 -20159 M ES -25334 -25417 In-Out -26142 -26225

3 states Tsallis Entropy 2 states Tsallis Entropy

(40)

39

crises. In this case it is difficult to identify the dot-com bubble; it could be related with the contagion effect that usually has strong changes during the financial crises. But also this systemic risk measure could be sensible to other situation that affects the connectedness of the system like wars or political events. In contrast with the case including the trend the expected duration of the regimes is shorter. For example the Expected duration of Regime #2: 170.79 time periods.

Figure 19: Smoothed states Probabilities for Shannon entropy based on In-Out

0 1000 2000 3000 4000 5000 6000 7000 8000 -2 0 2 Explained Variable #1 0 1000 2000 3000 4000 5000 6000 7000 8000 -0.5 0x 10 -14

0 1000 2000 3000 4000 5000 6000 7000 8000 0 0.5 1 Conditional Std of Equation #1 0 1000 2000 3000 4000 5000 6000 7000 8000 0 0.5 1 Time S m o o th e d S ta te s P ro b a b ili ti e s State 1 State 2 State 3

applying Markov Chain switching model to Systemic Risk measures

EntropyIndexentropyIndexentr

opyIndexentropyIndexentropyI

ndexertySystemicriskmeasures

ystemikrisksystemicriskmeasur

esystemicrisksystemicriskmeas

ureMESpasdfghjklzxcvbnmqwer

tyuiopasdfghjklzxcvbnmqwerty

uiopasdfghjklzxcvbnmqwertyui

opasdfghjklzxcvbnmqwertyuiop

entropyIndexentropyIndexentr

opyIndexentropyIndexentropyI

ndexertySystemicriskmeasuree

ntropyshannonrényientropytsal

lisentropyentropyIndexentropy

IndexentropyIndexentropyInde

xentropyIndexertySystemicrisk

measureentropyshannonrényie

Markov Chain Switching

model applied on entropy

index based on Systemic

Risk Measures

Acknowledgements

Abstract

Contents

Introduction

1. Systemic Risk

2. Systemic Risk Measures

3. Entropy index

4. Markov Chain Switching Regime

5. Estimation

6. Results

7. MSCM Removing the trend